In Class Ex5 - Spatially Weighted Regression

Overview

Packages used:

  • here: used to generate the file path to a specific folder
  • tidyverse
    • readr: for reading rectangular files
  • sf: Used to handle spatial dataframes
  • funModeling: Used for plotting and EDA
  • tmap: Use for plotting geo spatial maps
  • skimr: provides summary statistics about variables in the dataframe
  • corrplot: Used for performing correlation analysis
  • blorr: Used to generate a report for our logistic regression model
  • caret: Package for confusion matrix

Step 1: Load the required packages

# pacman::p_load(here,
#                sf, tidyverse, spdep,
#                funModeling, tmap, ggpubr,
#                corrplot, heatmaply, cluster, ClustGeo, factoextra, GGally,
#                blorr, skimr, caret, GWmodel,report
#                )

We load the packages as follow:

pacman::p_load(here, sf, tidyverse, funModeling, tmap,
               skimr, corrplot, blorr, GWmodel, caret
              )

We create the boundary path with the here() function:

boundary_path <- here("data", "dataOsun", "Osun.rds")
boundary_path
[1] "D:/f4sared/ISSS624/data/dataOsun/Osun.rds"

We create the path to the water point data:

data_path <- here("data", "dataOsun", "Osun_wp_sf.rds")
data_path
[1] "D:/f4sared/ISSS624/data/dataOsun/Osun_wp_sf.rds"

Using read_rds() from readr (tidyverse):

osun <- read_rds(boundary_path)

Using read_rds() from readr (tidyverse):

osun_wp_sf <- read_rds(data_path)

Using freq() from funModeling, we explore the status distribution:

osun_wp_sf %>% freq(input = 'status')
Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
of ggplot2 3.3.4.
ℹ The deprecated feature was likely used in the funModeling package.
  Please report the issue at <https://github.com/pablo14/funModeling/issues>.

  status frequency percentage cumulative_perc
1   TRUE      2642       55.5            55.5
2  FALSE      2118       44.5           100.0

Using the tmap package, we will plot the functional and non-functional water points:

tmap_mode('view')
tmap mode set to interactive viewing
tm_shape(osun)+
  tm_polygons(alpha=0.4)+
  tm_shape(osun_wp_sf) + 
  tm_dots(col="status", alpha = 0.6) + 
  tm_view(set.zoom.limits = c(9,12))

Step 2: EDA

Using the skim() from skimr package, we obtain the summary statistics as follow:

osun_wp_sf %>% skim() 
Warning: Couldn't find skimmers for class: sfc_POINT, sfc; No user-defined `sfl`
provided. Falling back to `character`.
Data summary
Name Piped data
Number of rows 4760
Number of columns 75
_______________________
Column type frequency:
character 47
logical 5
numeric 23
________________________
Group variables None

Variable type: character

skim_variable n_missing complete_rate min max empty n_unique whitespace
source 0 1.00 5 44 0 2 0
report_date 0 1.00 22 22 0 42 0
status_id 0 1.00 2 7 0 3 0
water_source_clean 0 1.00 8 22 0 3 0
water_source_category 0 1.00 4 6 0 2 0
water_tech_clean 24 0.99 9 23 0 3 0
water_tech_category 24 0.99 9 15 0 2 0
facility_type 0 1.00 8 8 0 1 0
clean_country_name 0 1.00 7 7 0 1 0
clean_adm1 0 1.00 3 5 0 5 0
clean_adm2 0 1.00 3 14 0 35 0
clean_adm3 4760 0.00 NA NA 0 0 0
clean_adm4 4760 0.00 NA NA 0 0 0
installer 4760 0.00 NA NA 0 0 0
management_clean 1573 0.67 5 37 0 7 0
status_clean 0 1.00 9 32 0 7 0
pay 0 1.00 2 39 0 7 0
fecal_coliform_presence 4760 0.00 NA NA 0 0 0
subjective_quality 0 1.00 18 20 0 4 0
activity_id 4757 0.00 36 36 0 3 0
scheme_id 4760 0.00 NA NA 0 0 0
wpdx_id 0 1.00 12 12 0 4760 0
notes 0 1.00 2 96 0 3502 0
orig_lnk 4757 0.00 84 84 0 1 0
photo_lnk 41 0.99 84 84 0 4719 0
country_id 0 1.00 2 2 0 1 0
data_lnk 0 1.00 79 96 0 2 0
water_point_history 0 1.00 142 834 0 4750 0
clean_country_id 0 1.00 3 3 0 1 0
country_name 0 1.00 7 7 0 1 0
water_source 0 1.00 8 30 0 4 0
water_tech 0 1.00 5 37 0 20 0
adm2 0 1.00 3 14 0 33 0
adm3 4760 0.00 NA NA 0 0 0
management 1573 0.67 5 47 0 7 0
adm1 0 1.00 4 5 0 4 0
New Georeferenced Column 0 1.00 16 35 0 4760 0
lat_lon_deg 0 1.00 13 32 0 4760 0
public_data_source 0 1.00 84 102 0 2 0
converted 0 1.00 53 53 0 1 0
created_timestamp 0 1.00 22 22 0 2 0
updated_timestamp 0 1.00 22 22 0 2 0
Geometry 0 1.00 33 37 0 4760 0
ADM2_EN 0 1.00 3 14 0 30 0
ADM2_PCODE 0 1.00 8 8 0 30 0
ADM1_EN 0 1.00 4 4 0 1 0
ADM1_PCODE 0 1.00 5 5 0 1 0

Variable type: logical

skim_variable n_missing complete_rate mean count
rehab_year 4760 0 NaN :
rehabilitator 4760 0 NaN :
is_urban 0 1 0.39 FAL: 2884, TRU: 1876
latest_record 0 1 1.00 TRU: 4760
status 0 1 0.56 TRU: 2642, FAL: 2118

Variable type: numeric

skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
row_id 0 1.00 68550.48 10216.94 49601.00 66874.75 68244.50 69562.25 471319.00 ▇▁▁▁▁
lat_deg 0 1.00 7.68 0.22 7.06 7.51 7.71 7.88 8.06 ▁▂▇▇▇
lon_deg 0 1.00 4.54 0.21 4.08 4.36 4.56 4.71 5.06 ▃▆▇▇▂
install_year 1144 0.76 2008.63 6.04 1917.00 2006.00 2010.00 2013.00 2015.00 ▁▁▁▁▇
fecal_coliform_value 4760 0.00 NaN NA NA NA NA NA NA
distance_to_primary_road 0 1.00 5021.53 5648.34 0.01 719.36 2972.78 7314.73 26909.86 ▇▂▁▁▁
distance_to_secondary_road 0 1.00 3750.47 3938.63 0.15 460.90 2554.25 5791.94 19559.48 ▇▃▁▁▁
distance_to_tertiary_road 0 1.00 1259.28 1680.04 0.02 121.25 521.77 1834.42 10966.27 ▇▂▁▁▁
distance_to_city 0 1.00 16663.99 10960.82 53.05 7930.75 15030.41 24255.75 47934.34 ▇▇▆▃▁
distance_to_town 0 1.00 16726.59 12452.65 30.00 6876.92 12204.53 27739.46 44020.64 ▇▅▃▃▂
rehab_priority 2654 0.44 489.33 1658.81 0.00 7.00 91.50 376.25 29697.00 ▇▁▁▁▁
water_point_population 4 1.00 513.58 1458.92 0.00 14.00 119.00 433.25 29697.00 ▇▁▁▁▁
local_population_1km 4 1.00 2727.16 4189.46 0.00 176.00 1032.00 3717.00 36118.00 ▇▁▁▁▁
crucialness_score 798 0.83 0.26 0.28 0.00 0.07 0.15 0.35 1.00 ▇▃▁▁▁
pressure_score 798 0.83 1.46 4.16 0.00 0.12 0.41 1.24 93.69 ▇▁▁▁▁
usage_capacity 0 1.00 560.74 338.46 300.00 300.00 300.00 1000.00 1000.00 ▇▁▁▁▅
days_since_report 0 1.00 2692.69 41.92 1483.00 2688.00 2693.00 2700.00 4645.00 ▁▇▁▁▁
staleness_score 0 1.00 42.80 0.58 23.13 42.70 42.79 42.86 62.66 ▁▁▇▁▁
location_id 0 1.00 235865.49 6657.60 23741.00 230638.75 236199.50 240061.25 267454.00 ▁▁▁▁▇
cluster_size 0 1.00 1.05 0.25 1.00 1.00 1.00 1.00 4.00 ▇▁▁▁▁
lat_deg_original 4760 0.00 NaN NA NA NA NA NA NA
lon_deg_original 4760 0.00 NaN NA NA NA NA NA NA
count 0 1.00 1.00 0.00 1.00 1.00 1.00 1.00 1.00 ▁▁▇▁▁

Step 3: Selecting variables

Next, we will filter away the rows with na under the columns that we are interest in. We will also recode the usage_capacity as factor instead of the numerical form. This is done in preparation for the logistic regression.

We run the code as follow:

osun_wp_sf_clean <- osun_wp_sf %>% 
  filter_at(vars(status,
                 distance_to_primary_road,
                 distance_to_secondary_road,
                 distance_to_tertiary_road,
                 distance_to_city,
                 distance_to_town,
                 water_point_population,
                 local_population_1km,
                 usage_capacity,
                 is_urban,
                 water_source_clean),
            all_vars(!is.na(.))) %>% 
  mutate(usage_capacity = as.factor(usage_capacity))

Next we select the columns that we want:

osun_wp <- osun_wp_sf_clean %>% select(c(7,35:39,42:43,46:47,57)) %>% st_set_geometry(NULL)

Step 4: Correlation Analysis:

Next, we perform correlation analysis using cor() function of corrplot:

cluster_vars.cor = cor(osun_wp[,2:7])
corrplot.mixed(cluster_vars.cor,
               lower="ellipse",
               upper = "number",
               tl.pos="lt",
               diag="l",
               tl.col="black")

From the corrplot above, we do not see any variables that are highly correlated. We can then proceed with building a simple logistic regression model

Step 5: Building a simple logistic regression model

Logistic Regression Model

Using the glm() of the R statistics, we build a logistic regression model as follow:

model <- glm(status ~ distance_to_primary_road +
               distance_to_secondary_road + 
               distance_to_tertiary_road + 
               distance_to_city + 
               distance_to_town + 
               is_urban +
               usage_capacity + 
               water_source_clean + 
               water_point_population + 
               local_population_1km,
             # data = osun_wp_sf_clean,
             data = osun_wp, 
             family = binomial(link = "logit"))

Model Report

Next we use blr_regress() of blorr to provide us a report on our model:

blr_regress(model)
                             Model Overview                              
------------------------------------------------------------------------
Data Set    Resp Var    Obs.    Df. Model    Df. Residual    Convergence 
------------------------------------------------------------------------
  data       status     4756      4755           4744           TRUE     
------------------------------------------------------------------------

                    Response Summary                     
--------------------------------------------------------
Outcome        Frequency        Outcome        Frequency 
--------------------------------------------------------
   0             2114              1             2642    
--------------------------------------------------------

                                 Maximum Likelihood Estimates                                   
-----------------------------------------------------------------------------------------------
               Parameter                    DF    Estimate    Std. Error    z value     Pr(>|z|) 
-----------------------------------------------------------------------------------------------
              (Intercept)                   1      0.3887        0.1124      3.4588       5e-04 
        distance_to_primary_road            1      0.0000        0.0000     -0.7153      0.4744 
       distance_to_secondary_road           1      0.0000        0.0000     -0.5530      0.5802 
       distance_to_tertiary_road            1      1e-04         0.0000      4.6708      0.0000 
            distance_to_city                1      0.0000        0.0000     -4.7574      0.0000 
            distance_to_town                1      0.0000        0.0000     -4.9170      0.0000 
              is_urbanTRUE                  1     -0.2971        0.0819     -3.6294       3e-04 
           usage_capacity1000               1     -0.6230        0.0697     -8.9366      0.0000 
water_source_cleanProtected Shallow Well    1      0.5040        0.0857      5.8783      0.0000 
   water_source_cleanProtected Spring       1      1.2882        0.4388      2.9359      0.0033 
         water_point_population             1      -5e-04        0.0000    -11.3686      0.0000 
          local_population_1km              1      3e-04         0.0000     19.2953      0.0000 
-----------------------------------------------------------------------------------------------

 Association of Predicted Probabilities and Observed Responses  
---------------------------------------------------------------
% Concordant          0.7347          Somers' D        0.4693   
% Discordant          0.2653          Gamma            0.4693   
% Tied                0.0000          Tau-a            0.2318   
Pairs                5585188          c                0.7347   
---------------------------------------------------------------

The report shows us that at 95% confidence level, all the variables are significant except distance to primary road and the distance to secondary road.

Note from class:

  • Categorical variable z value: +ve value implies above average correlation while -ve value implies a below average correlation.

  • Continuous variable z value: +ve value implies direct correlation while -ve value implies inverse correlation.

Confusion Matrix

Next, using blr_confusion_matrix() of blorr, we generate a confusion matrix as follow:

blr_confusion_matrix(model, cutoff=0.5)
Confusion Matrix and Statistics 

          Reference
Prediction FALSE TRUE
         0  1301  738
         1   813 1904

                Accuracy : 0.6739 
     No Information Rate : 0.4445 

                   Kappa : 0.3373 

McNemars's Test P-Value  : 0.0602 

             Sensitivity : 0.7207 
             Specificity : 0.6154 
          Pos Pred Value : 0.7008 
          Neg Pred Value : 0.6381 
              Prevalence : 0.5555 
          Detection Rate : 0.4003 
    Detection Prevalence : 0.5713 
       Balanced Accuracy : 0.6680 
               Precision : 0.7008 
                  Recall : 0.7207 

        'Positive' Class : 1

Notes from class:

The cutoff value of 0.5 means that any probability above 0.5 gets classified as true, while other values get classified as false. The default value we generally use is 0.5.

  • Sensitivity: TP/(TP/FN) >> 0.7207

  • Specificity: TN/(TN+FP) >> 0.6154

  • Accuracy: (TP+TN)/(TP+FP+TN+FN)

Since sensitivity is higher than the specificity, our model is better at catching true positives than catching true negatives.

Step 6: Spatially weighted geographical regression gwLR model

Converting to spatial data

Notes: In the past, spatial data used to be the common currency in model building. So many functions still take in spatial data. Hence the conversion is needed. The simple features dataframe only came in recently.

First we need to convert to spatial data as follow:

osun_wp_sp <- osun_wp_sf_clean %>% 
  select(c(status,
           distance_to_primary_road,
           distance_to_secondary_road,
           distance_to_tertiary_road,
           distance_to_city,
           distance_to_town,
           water_point_population,
           local_population_1km,
           is_urban,
           usage_capacity,
           water_source_clean)) %>% as_Spatial() 
osun_wp_sp
class       : SpatialPointsDataFrame 
features    : 4756 
extent      : 182502.4, 290751, 340054.1, 450905.3  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs 
variables   : 11
names       : status, distance_to_primary_road, distance_to_secondary_road, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, is_urban, usage_capacity, water_source_clean 
min values  :      0,        0.014461356813335,          0.152195902540837,         0.017815121653488, 53.0461399623541, 30.0019777713073,                      0,                    0,        0,           1000,           Borehole 
max values  :      1,         26909.8616132094,           19559.4793799085,          10966.2705628969,  47934.343603562, 44020.6393368124,                  29697,                36118,        1,            300,   Protected Spring 

Determine the fixed bandwidth

Using bw.ggwr() from GWmodel package, we will next determine the fixed bandwidth:

bw.fixed <- bw.ggwr(status ~ 
                      distance_to_primary_road + 
                      distance_to_secondary_road+
                      distance_to_tertiary_road +
                      distance_to_city+
                      distance_to_town+
                      water_point_population+
                      local_population_1km+
                      is_urban+
                      usage_capacity+
                      water_source_clean,
                    data=osun_wp_sp,
                    family="binomial",
                    approach="AIC",
                    kernel = "gaussian",
                    adaptive = FALSE,
                    longlat = FALSE)
bw.fixed

Creating the model

using ggwr.basic() from the GWmodel, we will build the model as follow:

gwlr.fixed <- ggwr.basic(status ~
                           distance_to_primary_road + 
                           distance_to_secondary_road + 
                           distance_to_tertiary_road + 
                           distance_to_city + 
                           distance_to_town + 
                           water_point_population + 
                           local_population_1km + 
                           is_urban + 
                           usage_capacity + 
                           water_source_clean,
                         data=osun_wp_sp,
                         bw=2599.672,
                         family = "binomial",
                         kernel = "gaussian",
                         adaptive = FALSE,
                         longlat = FALSE)
 Iteration    Log-Likelihood
=========================
       0        -1958 
       1        -1676 
       2        -1526 
       3        -1443 
       4        -1405 
       5        -1405 

We get the results as follow:

gwlr.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-18 11:35:57 
   Call:
   ggwr.basic(formula = status ~ distance_to_primary_road + distance_to_secondary_road + 
    distance_to_tertiary_road + distance_to_city + distance_to_town + 
    water_point_population + local_population_1km + is_urban + 
    usage_capacity + water_source_clean, data = osun_wp_sp, bw = 2599.672, 
    family = "binomial", kernel = "gaussian", adaptive = FALSE, 
    longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_primary_road distance_to_secondary_road distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km is_urban usage_capacity water_source_clean
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-124.555    -1.755     1.072     1.742    34.333  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.887e-01  1.124e-01   3.459 0.000543
distance_to_primary_road                 -4.642e-06  6.490e-06  -0.715 0.474422
distance_to_secondary_road               -5.143e-06  9.299e-06  -0.553 0.580230
distance_to_tertiary_road                 9.683e-05  2.073e-05   4.671 3.00e-06
distance_to_city                         -1.686e-05  3.544e-06  -4.757 1.96e-06
distance_to_town                         -1.480e-05  3.009e-06  -4.917 8.79e-07
water_point_population                   -5.097e-04  4.484e-05 -11.369  < 2e-16
local_population_1km                      3.451e-04  1.788e-05  19.295  < 2e-16
is_urbanTRUE                             -2.971e-01  8.185e-02  -3.629 0.000284
usage_capacity1000                       -6.230e-01  6.972e-02  -8.937  < 2e-16
water_source_cleanProtected Shallow Well  5.040e-01  8.574e-02   5.878 4.14e-09
water_source_cleanProtected Spring        1.288e+00  4.388e-01   2.936 0.003325
                                            
Intercept                                ***
distance_to_primary_road                    
distance_to_secondary_road                  
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
water_point_population                   ***
local_population_1km                     ***
is_urbanTRUE                             ***
usage_capacity1000                       ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.0  on 4744  degrees of freedom
AIC: 5712

Number of Fisher Scoring iterations: 5


 AICc:  5712.099
 Pseudo R-square value:  0.1295351
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2599.672 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -8.7229e+02 -4.9955e+00  1.7600e+00
   distance_to_primary_road                 -1.9389e-02 -4.8031e-04  2.9618e-05
   distance_to_secondary_road               -1.5921e-02 -3.7551e-04  1.2317e-04
   distance_to_tertiary_road                -1.5618e-02 -4.2368e-04  7.6179e-05
   distance_to_city                         -1.8416e-02 -5.6217e-04 -1.2726e-04
   distance_to_town                         -2.2411e-02 -5.7283e-04 -1.5155e-04
   water_point_population                   -5.2208e-02 -2.2767e-03 -9.8875e-04
   local_population_1km                     -1.2698e-01  4.9952e-04  1.0638e-03
   is_urbanTRUE                             -1.9790e+02 -4.2908e+00 -1.6864e+00
   usage_capacity1000                       -2.0772e+01 -9.7231e-01 -4.1592e-01
   water_source_cleanProtected.Shallow.Well -2.0789e+01 -4.5190e-01  5.3340e-01
   water_source_cleanProtected.Spring       -5.2235e+02 -5.5977e+00  2.5441e+00
                                                3rd Qu.      Max.
   Intercept                                 1.2763e+01 1073.2156
   distance_to_primary_road                  4.8443e-04    0.0142
   distance_to_secondary_road                6.0692e-04    0.0258
   distance_to_tertiary_road                 6.6815e-04    0.0128
   distance_to_city                          2.3718e-04    0.0150
   distance_to_town                          1.9271e-04    0.0224
   water_point_population                    5.0102e-04    0.1309
   local_population_1km                      1.8157e-03    0.0392
   is_urbanTRUE                              1.2841e+00  744.3099
   usage_capacity1000                        3.0322e-01    5.9281
   water_source_cleanProtected.Shallow.Well  1.7849e+00   67.6343
   water_source_cleanProtected.Spring        6.7663e+00  317.4133
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 2795.084 
   AIC : 4414.606 
   AICc : 4747.423 
   Pseudo R-square value:  0.5722559 

   ***********************************************************************
   Program stops at: 2022-12-18 11:36:46 

A lower AIC indicated a better fit model.

Confusion Matrix

We will convert the SDF object to a dataframe first:

gwr.fixed <- as.data.frame(gwlr.fixed$SDF)

Next, we will change the probability to True or False:

gwr.fixed <- gwr.fixed %>% mutate(most=ifelse(gwr.fixed$yhat >= 0.5, T, F))

Using confusionMatrix() of the caret package we run the following code:

gwr.fixed$y <- as.factor(gwr.fixed$y)
gwr.fixed$most <- as.factor(gwr.fixed$most)
CM <- confusionMatrix(data=gwr.fixed$most, reference = gwr.fixed$y, positive = "TRUE")
CM
Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1824  263
     TRUE    290 2379
                                          
               Accuracy : 0.8837          
                 95% CI : (0.8743, 0.8927)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7642          
                                          
 Mcnemar's Test P-Value : 0.2689          
                                          
            Sensitivity : 0.9005          
            Specificity : 0.8628          
         Pos Pred Value : 0.8913          
         Neg Pred Value : 0.8740          
             Prevalence : 0.5555          
         Detection Rate : 0.5002          
   Detection Prevalence : 0.5612          
      Balanced Accuracy : 0.8816          
                                          
       'Positive' Class : TRUE            
                                          

Here we see a big improvement in accuracy, sensitivity & specificity. Thus by incorporating the geographical weights, we have now obtained a better model with better excitability.

Step 7: Visualizing the results

First we will select the required columns:

osun_wp_sf_selected <- osun_wp_sf_clean %>% select(c(ADM2_EN, ADM2_PCODE,
                                                     ADM1_EN, ADM1_PCODE,
                                                     status))

Next we will cbind the earlier model results to get a new dataframe for plotting:

gwr_sf.fixed <- cbind(osun_wp_sf_selected, gwr.fixed)

We plot the map as follow:

tmap_mode("view")
tmap mode set to interactive viewing
prob_T <- tm_shape(osun) + 
  tm_polygons(alpha=0.1) + 
  tm_shape(gwr_sf.fixed) + 
  tm_dots(col="yhat",
          border.col = "gray60",
          border.lwd = 1) + 
  tm_view(set.zoom.limits = c(9,14))

actual <- tm_shape(osun) + 
  tm_polygons(alpha=0.1) + 
  tm_shape(gwr_sf.fixed) + 
  tm_dots(col="status",
          border.col = "gray60",
          border.lwd = 1) + 
  tm_view(set.zoom.limits = c(9,14))

tmap_arrange(actual, prob_T, 
             asp = 1, ncol = 2, sync = TRUE)

Show only false results

We perform a simple filter:

false_results <- gwr_sf.fixed %>% filter(status == "FALSE")

Next we plot as normal:

tmap_mode("view")
tmap mode set to interactive viewing
prob_T_false <- tm_shape(osun) + 
  tm_polygons(alpha=0.1) + 
  tm_shape(false_results) + 
  tm_dots(col="yhat",
          border.col = "gray60",
          border.lwd = 1) + 
  tm_view(set.zoom.limits = c(9,14))

actual <- tm_shape(osun) + 
  tm_polygons(alpha=0.1) + 
  tm_shape(false_results) + 
  tm_dots(col="status",
          border.col = "gray60",
          border.lwd = 1) + 
  tm_view(set.zoom.limits = c(9,14))

tmap_arrange(actual, prob_T_false, 
             asp = 1, ncol = 2, sync = TRUE)

Here we can see that among the false results, there are quite a few false negatives.

Step 8: gwlr with only statistically significant variables

Determine new fixed bandwidth

bw.fixed_new <- bw.ggwr(status ~ 
#                     distance_to_primary_road + 
#                     distance_to_secondary_road+
                      distance_to_tertiary_road +
                      distance_to_city+
                      distance_to_town+
                      water_point_population+
                      local_population_1km+
                      is_urban+
                      usage_capacity+
                      water_source_clean,
                    data=osun_wp_sp,
                    family="binomial",
                    approach="AIC",
                    kernel = "gaussian",
                    adaptive = FALSE,
                    longlat = FALSE)
bw.fixed_new

Creating the new model

Using the new bandwidth, we create the new model

gwlr.fixed_new <- ggwr.basic(status ~
                           distance_to_tertiary_road + 
                           distance_to_city + 
                           distance_to_town + 
                           water_point_population + 
                           local_population_1km + 
                           is_urban + 
                           usage_capacity + 
                           water_source_clean,
                         data=osun_wp_sp,
                         bw=2377.371,
                         family = "binomial",
                         kernel = "gaussian",
                         adaptive = FALSE,
                         longlat = FALSE)
 Iteration    Log-Likelihood
=========================
       0        -1959 
       1        -1680 
       2        -1531 
       3        -1447 
       4        -1413 
       5        -1413 
gwlr.fixed_new
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-18 11:36:49 
   Call:
   ggwr.basic(formula = status ~ distance_to_tertiary_road + distance_to_city + 
    distance_to_town + water_point_population + local_population_1km + 
    is_urban + usage_capacity + water_source_clean, data = osun_wp_sp, 
    bw = 2377.371, family = "binomial", kernel = "gaussian", 
    adaptive = FALSE, longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km is_urban usage_capacity water_source_clean
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-129.368    -1.750     1.074     1.742    34.126  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.540e-01  1.055e-01   3.354 0.000796
distance_to_tertiary_road                 1.001e-04  2.040e-05   4.910 9.13e-07
distance_to_city                         -1.764e-05  3.391e-06  -5.202 1.97e-07
distance_to_town                         -1.544e-05  2.825e-06  -5.466 4.60e-08
water_point_population                   -5.098e-04  4.476e-05 -11.390  < 2e-16
local_population_1km                      3.452e-04  1.779e-05  19.407  < 2e-16
is_urbanTRUE                             -2.667e-01  7.474e-02  -3.569 0.000358
usage_capacity1000                       -6.206e-01  6.966e-02  -8.908  < 2e-16
water_source_cleanProtected Shallow Well  4.947e-01  8.496e-02   5.823 5.79e-09
water_source_cleanProtected Spring        1.279e+00  4.384e-01   2.917 0.003530
                                            
Intercept                                ***
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
water_point_population                   ***
local_population_1km                     ***
is_urbanTRUE                             ***
usage_capacity1000                       ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.9  on 4746  degrees of freedom
AIC: 5708.9

Number of Fisher Scoring iterations: 5


 AICc:  5708.923
 Pseudo R-square value:  0.129406
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2377.371 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -3.7021e+02 -4.3797e+00  3.5590e+00
   distance_to_tertiary_road                -3.1622e-02 -4.5462e-04  9.1291e-05
   distance_to_city                         -5.4555e-02 -6.5623e-04 -1.3507e-04
   distance_to_town                         -8.6549e-03 -5.2754e-04 -1.6785e-04
   water_point_population                   -2.9696e-02 -2.2705e-03 -1.2277e-03
   local_population_1km                     -7.7730e-02  4.4281e-04  1.0548e-03
   is_urbanTRUE                             -7.3554e+02 -3.4675e+00 -1.6596e+00
   usage_capacity1000                       -5.5889e+01 -1.0347e+00 -4.1960e-01
   water_source_cleanProtected.Shallow.Well -1.8842e+02 -4.7295e-01  6.2378e-01
   water_source_cleanProtected.Spring       -1.3630e+03 -5.3436e+00  2.7714e+00
                                                3rd Qu.      Max.
   Intercept                                 1.3755e+01 2171.6375
   distance_to_tertiary_road                 6.3011e-04    0.0237
   distance_to_city                          1.5921e-04    0.0162
   distance_to_town                          2.4490e-04    0.0179
   water_point_population                    4.5879e-04    0.0765
   local_population_1km                      1.8479e-03    0.0333
   is_urbanTRUE                              1.0554e+00  995.1841
   usage_capacity1000                        3.9113e-01    9.2449
   water_source_cleanProtected.Shallow.Well  1.9564e+00   66.8914
   water_source_cleanProtected.Spring        7.0805e+00  208.3749
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 2815.659 
   AIC : 4418.776 
   AICc : 4744.213 
   Pseudo R-square value:  0.5691072 

   ***********************************************************************
   Program stops at: 2022-12-18 11:37:30 

Confusion Matrix

Next we will make a dataframe from the earlier created model:

gwlr.fixed_new <- as.data.frame(gwlr.fixed_new$SDF)

We will then assign TRUE or FALSE based on the output probability:

gwr.fixed_new <- gwlr.fixed_new %>% mutate(most=ifelse(gwlr.fixed_new$yhat >= 0.5, T, F))

Then we can plot the confusion matrix as follow:

gwr.fixed_new$y <- as.factor(gwr.fixed_new$y)
gwr.fixed_new$most <- as.factor(gwr.fixed_new$most)
CM <- confusionMatrix(data=gwr.fixed_new$most, reference = gwr.fixed_new$y, positive = "TRUE")
CM
Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1833  268
     TRUE    281 2374
                                          
               Accuracy : 0.8846          
                 95% CI : (0.8751, 0.8935)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7661          
                                          
 Mcnemar's Test P-Value : 0.6085          
                                          
            Sensitivity : 0.8986          
            Specificity : 0.8671          
         Pos Pred Value : 0.8942          
         Neg Pred Value : 0.8724          
             Prevalence : 0.5555          
         Detection Rate : 0.4992          
   Detection Prevalence : 0.5582          
      Balanced Accuracy : 0.8828          
                                          
       'Positive' Class : TRUE            
                                          

From the results, we can see that by taking geographical relationships into account, we are then able to obtain a better model with higher accuracy. The improvement in the models ability to explain is also shown in the improvement in the sensitivity and specificity values.

Step 9: Visualizing the new results

Let’s then combine the earlier results with the region boundary data in order to plot:

gwr_sf.fixed_new <- cbind(osun_wp_sf_selected, gwr.fixed_new)
tmap_mode("view")
tmap mode set to interactive viewing
prob_T_new <- tm_shape(osun) + 
  tm_polygons(alpha=0.1) + 
  tm_shape(gwr_sf.fixed_new) + 
  tm_dots(col="yhat",
          border.col = "gray60",
          border.lwd = 1) + 
  tm_view(set.zoom.limits = c(9,14))

actual <- tm_shape(osun) + 
  tm_polygons(alpha=0.1) + 
  tm_shape(gwr_sf.fixed_new) + 
  tm_dots(col="status",
          border.col = "gray60",
          border.lwd = 1) + 
  tm_view(set.zoom.limits = c(9,14))

tmap_arrange(actual, prob_T_new, 
             asp = 1, ncol = 2, sync = TRUE)

Show only false results

We perform a simple filter for only false results

false_results_new <- gwr_sf.fixed_new %>% filter(status == "FALSE")

Next we visualize the plot:

tmap_mode("view")
tmap mode set to interactive viewing
prob_T_false_new <- tm_shape(osun) + 
  tm_polygons(alpha=0.1) + 
  tm_shape(false_results_new) + 
  tm_dots(col="yhat",
          border.col = "gray60",
          border.lwd = 1) + 
  tm_view(set.zoom.limits = c(9,14))

tmap_arrange(prob_T_false, prob_T_false_new, 
             asp = 1, ncol = 2, sync = TRUE)

In the above plot, only the false results are plotted. Since all are false results, in the ideal world, we should only be seeing the white color dots instead of the darker color dots. The left plot is the original prediction from the earlier model while the right plot is the result from the new models with insignificant variables removed.

We see that for certain water points on the right plot, the color intensity actually decreased which shows the improvement of the model’s explain ability.

However, it is clear that for certain waterpoints, the false positive is still high. This means that additional data may still be needed to fully explain the results.